Symbolic computation for differential equation based systems

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This project is in the area of symbolic computational dynamics. This is a new approach to the modelling and solution of differential equation-based problems in science and engineering. The academic roots of symbolic computational dynamics can be traced back to some early implementations of semi-automated differential equation solution code written in the MACSYMA language in the seventies. Since then a lot of research work has been done in this area by the applicant and others working with him, and the approach which we now call symbolic computational dynamics has been evolved by us over the last 25 years. A significant proportion of that work was funded by EPSRC (Cartmell PI) A start-up company, Symbolic Computational Dynamics Ltd (SCD Ltd) was incorporated in Glasgow in June 2020 to drive the product development work. SCD Ltd is funding this studentship at a level of £12680 because there is a parallel and vital strand of academic work to be done which is needed to support the product development activity being undertaken in the Company.The topic of Symbolic Computational Dynamics is based on the fact that all real-world dynamical systems are mathematically nonlinear, and that modelling simplifications through scaling down and linearisation can only ever be of limited use because usually only the lowest order phenomena are captured that way. Finite Element Analysis potentially offers a versatile and powerful way round this problem but is limited in generality by the specific numerical representation of the problem upon which the FEA model is based. This means that FEA becomes computationally expensive if a global understanding of the problem domain is required. Symbolic Computational Dynamics is based entirely on differential equations, either systems of ordinary nonlinear differential equations, or systems of partial nonlinear differential equations and boundary conditions, and constructing approximate analytical solutions to these equations using asymptotic analysis. Traditionally this sort of nonlinear asymptotic analysis is done by hand and for very small systems of differential equations, because of the algebraic complexity and the sheer effort needed. Now this can all be done on a computer, running the computational solver codes that have been developed by the team over the last 25 years, and with different levels of accuracy as required. The latest version of this suite of solvers is capable of doing an entire asymptotic analysis to second or even higher order perturbation approximation in a matter of a few minutes. This is achieved by running un-optimised serial code on a cloud-based server application, so speed-up techniques will be researched. In addition to this we will build a new generation of user interfaces (UIs) to enable interaction with the user.